Expected Value Ontology

flow "ExpectedValueOntology" v4.0

# EXPECTED VALUE ONTOLOGY
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/risk-decision-mathematics/expected-value-ontology.html"
  title: "Expected Value Ontology"
  domain: "Probability and Statistics"
  category: "Risk & Decision Mathematics"
  entity: EXPECTED_VALUE
  entity_type: PROBABILITY_DECISION_ONTOLOGY

DEFINE:
  EXPECTED_VALUE :=
    PROBABILITY_WEIGHTED_AVERAGE(
      RANDOM_VARIABLE,
      POSSIBLE_VALUES,
      PROBABILITY_DISTRIBUTION
    )

PURPOSE:
  REPRESENT:
    LONG_RUN_AVERAGE
    DISTRIBUTION_CENTRE
    PROBABILITY_WEIGHTED_OUTCOME
    EXPECTED_GAIN
    EXPECTED_LOSS
    EXPECTED_UTILITY

FOUNDATION:
  RANDOM_VARIABLE: X
  DISCRETE_EXPECTATION:
    E[X] := SUM_x x P(X=x)
  CONTINUOUS_EXPECTATION:
    E[X] := INTEGRAL x f_X(x) dx
  GENERAL_EXPECTATION:
    E[X] := INTEGRAL X dP

ENTITIES:
  EXPECTED_VALUE: PROBABILITY_WEIGHTED_MEAN
  EXPECTATION: EXPECTED_VALUE_OPERATOR
  RANDOM_VARIABLE: MEASURABLE_NUMERICAL_OUTCOME
  PROBABILITY_DISTRIBUTION: UNCERTAINTY_STRUCTURE
  MEAN: DISTRIBUTIONAL_CENTRE
  EXPECTED_GAIN: EXPECTED_POSITIVE_VALUE
  EXPECTED_LOSS: EXPECTED_COST_OR_DAMAGE
  EXPECTED_UTILITY: EXPECTED_PREFERENCE_VALUE
  CONDITIONAL_EXPECTATION: EXPECTATION_GIVEN_INFORMATION

RELATIONSHIPS:
  RANDOM_VARIABLE -> HAS -> PROBABILITY_DISTRIBUTION
  PROBABILITY_DISTRIBUTION -> WEIGHTS -> POSSIBLE_VALUES
  WEIGHTED_VALUES -> AGGREGATE_TO -> EXPECTED_VALUE
  EXPECTED_VALUE -> CONNECTS_TO -> MEAN
  EXPECTED_VALUE -> SUPPORTS -> DECISION_MAKING
  CONDITIONAL_INFORMATION -> MODIFIES -> EXPECTATION
  EXPECTED_LOSS -> SUPPORTS -> RISK_ANALYSIS
  EXPECTED_UTILITY -> SUPPORTS -> DECISION_THEORY

DISCRETE_EXPECTED_VALUE:
  values: x_1,...,x_n
  probabilities: p_1,...,p_n
  requirements:
    p_i >= 0
    SUM_i p_i = 1
  expectation:
    E[X] := SUM_i x_i p_i

CONTINUOUS_EXPECTED_VALUE:
  density: f_X(x)
  requirements:
    f_X(x) >= 0
    INTEGRAL f_X(x) dx = 1
  expectation:
    E[X] := INTEGRAL x f_X(x) dx

EXISTENCE:
  sufficient_condition:
    E[|X|] < infinity
  caution:
    SOME_DISTRIBUTIONS_DO_NOT_HAVE_FINITE_EXPECTED_VALUE

LINEARITY_OF_EXPECTATION:
  property:
    E[aX+bY] = aE[X]+bE[Y]
  independence:
    NOT_REQUIRED

CONSTANT_EXPECTATION:
  property:
    E[c] = c

SUM_EXPECTATION:
  property:
    E[SUM_i X_i] = SUM_i E[X_i]
  independence:
    NOT_REQUIRED

PRODUCT_EXPECTATION:
  IF:
    X_AND_Y_INDEPENDENT
  THEN:
    E[XY] = E[X]E[Y]
  caution:
    NOT_VALID_FOR_ARBITRARY_DEPENDENT_VARIABLES

FUNCTION_OF_RANDOM_VARIABLE:
  transformation:
    Y := g(X)
  expectation:
    E[g(X)]

LOTUS:
  full_name:
    LAW_OF_THE_UNCONSCIOUS_STATISTICIAN
  role:
    COMPUTE_EXPECTATION_OF_FUNCTION_WITHOUT_DERIVING_TRANSFORMED_DISTRIBUTION

INDICATOR_EXPECTATION:
  property:
    E[I_A] = P(A)

BERNOULLI_EXPECTATION:
  X ~ Bernoulli(p)
  expectation:
    E[X] = p

BINOMIAL_EXPECTATION:
  X ~ Binomial(n,p)
  expectation:
    E[X] = np

POISSON_EXPECTATION:
  X ~ Poisson(lambda)
  expectation:
    E[X] = lambda

NORMAL_EXPECTATION:
  X ~ Normal(mu,sigma_squared)
  expectation:
    E[X] = mu

EXPONENTIAL_EXPECTATION:
  X ~ Exponential(lambda)
  expectation:
    E[X] = 1/lambda
  parameterisation:
    RATE_lambda

EXPECTED_VALUE_VS_MEDIAN:
  rule:
    EXPECTED_VALUE != MEDIAN_BY_DEFAULT

EXPECTED_VALUE_VS_MODE:
  rule:
    EXPECTED_VALUE != MODE_BY_DEFAULT

EXPECTED_VALUE_VS_OBSERVED_VALUE:
  rule:
    EXPECTED_VALUE != GUARANTEED_OUTCOME

LONG_RUN_INTERPRETATION:
  repeated_trials:
    SAMPLE_MEAN_MAY_CONVERGE_TO_EXPECTED_VALUE
    UNDER:
      APPROPRIATE_LAW_OF_LARGE_NUMBERS_CONDITIONS

CONDITIONAL_EXPECTATION:
  notation:
    E[X|Y]
  meaning:
    EXPECTED_VALUE_OF_X_GIVEN_INFORMATION_ABOUT_Y
  property:
    E[X|Y]_IS_FUNCTION_OF_Y

LAW_OF_TOTAL_EXPECTATION:
  property:
    E[X] = E[E[X|Y]]

ITERATED_EXPECTATION:
  property:
    E[E[X|H]|G] = E[X|G]
    WHEN:
      G subseteq H

CONDITIONAL_EXPECTATION_GIVEN_EVENT:
  requirement:
    P(A) > 0
  property:
    E[X|A] = E[X I_A]/P(A)

EXPECTED_GAIN:
  expected_gain:
    E[G]

EXPECTED_LOSS:
  expected_loss:
    E[L]
  decision_rule:
    SELECT_ACTION_MINIMISING_EXPECTED_LOSS
    UNDER:
      RISK_NEUTRAL_EXPECTED_LOSS_CRITERION

EXPECTED_UTILITY:
  expected_utility:
    E[U(X)]
  decision_rule:
    SELECT_ACTION_MAXIMISING_EXPECTED_UTILITY
    UNDER:
      EXPECTED_UTILITY_MODEL

EXPECTED_MONETARY_VALUE:
  alias: EMV
  EMV(a) :=
    SUM_i P(s_i)V(a,s_i)
  use:
    RISK_NEUTRAL_DECISION_ANALYSIS

DECISION_THEORY_CONNECTION:
  expected_loss:
    E[L(theta,a)]
  expected_utility:
    E[U(theta,a)]
  optimal_action:
    ARGMIN_EXPECTED_LOSS
    OR:
      ARGMAX_EXPECTED_UTILITY

RISK_ANALYSIS_CONNECTION:
  expected_loss:
    E[L]
  caution:
    EXPECTED_LOSS_ALONE_DOES_NOT_CAPTURE_TAIL_RISK_VARIABILITY_OR_WORST_CASE_LOSS

VARIANCE_CONNECTION:
  Var(X) :=
    E[(X-E[X])^2]
  equivalent:
    Var(X) = E[X^2]-E[X]^2
    WHEN:
      SECOND_MOMENT_FINITE

COVARIANCE_CONNECTION:
  Cov(X,Y) :=
    E[XY]-E[X]E[Y]

MOMENTS:
  raw_moment_k:
    E[X^k]
  central_moment_k:
    E[(X-E[X])^k]
  first_raw_moment:
    EXPECTED_VALUE

SAMPLE_MEAN:
  X_bar :=
    (1/n)*SUM_i X_i
  expectation:
    E[X_bar] = E[X_i]
    WHEN:
      X_i_HAVE_COMMON_FINITE_MEAN

UNBIASED_ESTIMATOR:
  condition:
    E_theta[T] = theta
  connection:
    EXPECTATION_DEFINES_UNBIASEDNESS

BIAS:
  Bias(T) :=
    E[T]-theta

MEAN_SQUARED_ERROR:
  MSE(T) :=
    E[(T-theta)^2]
  decomposition:
    MSE(T) = Var(T)+Bias(T)^2

JENSENS_INEQUALITY:
  convex_function:
    phi
  property:
    phi(E[X]) <= E[phi(X)]
    UNDER:
      APPROPRIATE_INTEGRABILITY
  connection:
    RISK_AVERSION
    INFORMATION_THEORY
    OPTIMISATION

MARKOV_INEQUALITY:
  requirements:
    X >= 0
    a > 0
  property:
    P(X >= a) <= E[X]/a

CHEBYSHEV_CONNECTION:
  property:
    P(|X-E[X]| >= k)
      <=
    Var(X)/k^2

MONTE_CARLO_ESTIMATION:
  samples:
    X_1,...,X_n
  estimator:
    E_hat[X] :=
      (1/n)*SUM_i X_i
  convergence:
    APPROACHES_EXPECTED_VALUE
    UNDER:
      LAW_OF_LARGE_NUMBERS_CONDITIONS

IMPORTANCE_SAMPLING_CONNECTION:
  target_distribution:
    p(x)
  proposal:
    q(x)
  expectation:
    E_p[f(X)]
      =
    E_q[
      f(X)p(X)/q(X)
    ]
    WHEN:
      SUPPORT_AND_INTEGRABILITY_CONDITIONS_HOLD

EXPECTED_VALUE_OF_INFORMATION:
  baseline:
    OPTIMAL_EXPECTED_VALUE_WITH_CURRENT_INFORMATION
  informed:
    OPTIMAL_EXPECTED_VALUE_WITH_ADDITIONAL_INFORMATION
  VOI:
    informed-baseline

EXPECTED_VALUE_OF_PERFECT_INFORMATION:
  alias: EVPI
  definition:
    EXPECTED_VALUE_WITH_PERFECT_INFORMATION
      -
    EXPECTED_VALUE_WITH_CURRENT_INFORMATION

EXPECTED_VALUE_OF_SAMPLE_INFORMATION:
  alias: EVSI
  definition:
    EXPECTED_VALUE_WITH_SAMPLE_INFORMATION
      -
    EXPECTED_VALUE_WITH_CURRENT_INFORMATION

PORTFOLIO_EXPECTED_RETURN:
  portfolio_return:
    R_p := SUM_i w_i R_i
  expectation:
    E[R_p] = SUM_i w_i E[R_i]
  independence:
    NOT_REQUIRED

STOCHASTIC_PROCESS_CONNECTION:
  process:
    X_t
  expected_value_function:
    m(t) := E[X_t]

MARTINGALE_CONNECTION:
  property:
    E[X_next|F_t] = X_t
    UNDER:
      MARTINGALE_CONDITIONS

APPLICATIONS:
  EXPECTED_VALUE -> SUPPORTS -> PROBABILITY_THEORY
  EXPECTED_VALUE -> SUPPORTS -> STATISTICS
  EXPECTED_VALUE -> SUPPORTS -> DECISION_THEORY
  EXPECTED_VALUE -> SUPPORTS -> RISK_ANALYSIS
  EXPECTED_VALUE -> SUPPORTS -> FINANCE
  EXPECTED_VALUE -> SUPPORTS -> STOCHASTIC_PROCESSES
  EXPECTED_VALUE -> SUPPORTS -> MACHINE_LEARNING
  EXPECTED_VALUE -> SUPPORTS -> MONTE_CARLO_SIMULATION

MODEL_CONNECTIONS:
  EXPECTED_VALUE -> CONNECTS_TO -> RANDOM_VARIABLE
  EXPECTED_VALUE -> CONNECTS_TO -> PROBABILITY_DISTRIBUTION
  EXPECTED_VALUE -> CONNECTS_TO -> VARIANCE
  EXPECTED_VALUE -> CONNECTS_TO -> COVARIANCE
  EXPECTED_VALUE -> CONNECTS_TO -> CONDITIONAL_EXPECTATION
  EXPECTED_VALUE -> CONNECTS_TO -> EXPECTED_UTILITY
  EXPECTED_VALUE -> CONNECTS_TO -> RISK_ANALYSIS
  EXPECTED_VALUE -> CONNECTS_TO -> DECISION_THEORY

SEMANTIC_EQUIVALENCE:
  "expected value"
    ~= "expectation"
    ~= "probability-weighted mean"

  "expected monetary value"
    ~= "probability-weighted monetary outcome"

  "conditional expectation"
    ~= "expected value given specified information"

  EXCLUDE_EQUIVALENCE:
    EXPECTED_VALUE != GUARANTEED_VALUE
    EXPECTED_VALUE != MEDIAN_BY_DEFAULT
    EXPECTED_VALUE != MODE_BY_DEFAULT
    EXPECTED_VALUE != SINGLE_OBSERVATION
    EXPECTED_LOSS != TAIL_RISK
    EXPECTED_VALUE != UTILITY
    SAMPLE_MEAN != POPULATION_EXPECTATION_BY_IDENTITY

VALIDATION_RULES:
  RULE_1:
    RANDOM_VARIABLE_AND_PROBABILITY_DISTRIBUTION_MUST_BE_DEFINED

  RULE_2:
    PROBABILITIES_MUST_BE_VALID_AND_NORMALISED

  RULE_3:
    EXPECTATION_EXISTENCE_OR_INTEGRABILITY_MUST_BE_CHECKED_WHEN_REQUIRED

  RULE_4:
    DISCRETE_CONTINUOUS_AND_GENERAL_EXPECTATION_FORMS_MUST_NOT_BE_CONFLATED

  RULE_5:
    INDEPENDENCE_MUST_NOT_BE_ASSUMED_FOR_LINEARITY_OF_EXPECTATION

  RULE_6:
    PRODUCT_EXPECTATION_FACTORIZATION_REQUIRES_APPROPRIATE_INDEPENDENCE_OR_JUSTIFICATION

  RULE_7:
    EXPECTED_VALUE_MUST_NOT_BE_INTERPRETED_AS_GUARANTEED_OR_MOST_LIKELY_OUTCOME

  RULE_8:
    EXPECTED_LOSS_MUST_BE_SUPPLEMENTED_WITH_OTHER_RISK_MEASURES_WHEN_TAIL_OR_VARIABILITY_MATTERS

CONFIDENCE:
  ontology_identity: 0.99
  discrete_expectation_structure: 0.99
  continuous_expectation_structure: 0.99
  linearity_structure: 0.99
  conditional_expectation_structure: 0.99
  decision_theory_relationship: 0.99
  semantic_distinctions: 0.99

UPDATE_MODEL:
  IF:
    PROBABILITY_DISTRIBUTION_CHANGES
  THEN:
    UPDATE:
      EXPECTED_VALUE
      CONDITIONAL_EXPECTATIONS
      MOMENTS
      EXPECTED_LOSSES
      EXPECTED_UTILITIES

  IF:
    NEW_EVIDENCE_ARRIVES
  THEN:
    UPDATE:
      CONDITIONAL_DISTRIBUTIONS
      CONDITIONAL_EXPECTATIONS
      DECISION_EXPECTATIONS

  IF:
    OUTCOME_OR_UTILITY_FUNCTION_CHANGES
  THEN:
    RECOMPUTE:
      EXPECTED_GAIN
      EXPECTED_LOSS
      EXPECTED_UTILITY
      VALUE_OF_INFORMATION

  PRESERVE:
    RANDOM_VARIABLE_DEFINITION
    SUPPORT
    PROBABILITY_DISTRIBUTION
    PARAMETERISATION
    OUTCOME_UNITS
    CONDITIONING_INFORMATION
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_RANDOM_VARIABLE
    - IDENTIFY_SUPPORT_AND_DISTRIBUTION
    - VERIFY_PROBABILITY_NORMALISATION
    - VERIFY_EXPECTATION_EXISTENCE
    - SELECT_DISCRETE_CONTINUOUS_OR_GENERAL_FORM
    - COMPUTE_EXPECTED_VALUE
    - COMPUTE_CONDITIONAL_EXPECTATION_IF_INFORMATION_PRESENT
    - APPLY_LINEARITY_WHEN_USEFUL
    - CONNECT_EXPECTATION_TO_VARIANCE_RISK_OR_DECISION_CONTEXT
    - RETURN_EXPECTED_VALUE_INTERPRETATION

  constraints:
    - PRESERVE_RANDOM_VARIABLE_SEMANTICS
    - PRESERVE_PROBABILITY_NORMALISATION
    - CHECK_INTEGRABILITY
    - DISTINGUISH_EXPECTED_VALUE_FROM_MEDIAN_AND_MODE
    - DO_NOT_REQUIRE_INDEPENDENCE_FOR_LINEARITY
    - REQUIRE_JUSTIFICATION_FOR_PRODUCT_FACTORIZATION
    - DISTINGUISH_EXPECTED_VALUE_FROM_GUARANTEED_OUTCOME
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "expected-value"
  namespace: "probability-statistics/risk-decision-mathematics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - PROBABILITY_WEIGHTED_AVERAGE
    - LINEARITY_OF_EXPECTATION
    - CONDITIONAL_EXPECTATION
    - DECISION_AND_RISK_CONNECTIONS

END_FLOW